A Closed-Form Value for the Generalized Uncertainty Principle
Parameter:
β
0
=
4
3
ln 2
from a Close-Packed Vacuum Lattice
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
August 21, 2026
Abstract
The generalized uncertainty principle (GUP) modies the Heisenberg relation by a term
β
0
(ℓ
P
∆p/ℏ)
2
and implies a minimal length
∆x
min
=
√
β
0
ℓ
P
. Theory expects
β
0
∼ 1
but
does not x its value; experiments bound it from above. Here we report a closed-form value,
β
0
=
4
3
ln 2 ≈ 0.9242
, obtained from two published inputs of the SelectionStitch Model (SSM): a
minimal approach distance
r
min
= L/
√
3
set by lattice geometry, and a bond length
L = 2
√
ln 2 ℓ
P
set by an exact, codeword-independent entanglement calibration of the model's
[[3L
3
s
, 2L
3
s
+2, 3]]
vacuum code. The same inputs x the model's macroscopic decoherence threshold
m
soft
= ℏ/(Lc)
,
which yields a bridge identity,
β
0
=
1
3
(m
P
/m
soft
)
2
. The identity ties two separate experimental
programs deformed-commutator tests and microgram-scale superposition tests to one quan-
tity, and it can be tested without adopting the model: a measured decoherence threshold predicts
β
0
, and conversely. All assumptions are tagged.
1 Introduction
The parameter.
Many approaches to quantum gravity modify the uncertainty relation at the Planck
scale [3, 4, 5, 7]. A standard form is the deformed commutator of Kempf, Mangano, and Mann [6],
[ˆx, ˆp] = iℏ
"
1 + β
0
ℓ
P
ˆp
ℏ
2
#
, ∆x ∆p ≥
ℏ
2
"
1 + β
0
ℓ
P
∆p
ℏ
2
#
.
(1)
The new term implies a smallest resolvable length,
∆x
min
=
√
β
0
ℓ
P
. The dimensionless constant
β
0
carries the physics.
The gap this paper addresses.
Theory expects
β
0
of order one, but no framework xes its value.
Experiments therefore bound it from above. Bounds come from gravitational bar detectors [9], macro-
scopic harmonic oscillators [10], and quantum-gravity corrections to standard quantum systems [8].
Depending on the system and on assumptions about how the deformation acts on composite objects,
the bounds span many orders of magnitude, from roughly
10
6
upward. None reaches
β
0
∼ 1
. A
denite predicted value would give these programs a target rather than a ceiling.
The claim.
We derive one:
β
0
=
4
3
ln 2 = 0.92420 . . .
(2)
1
The derivation uses two published inputs of the SelectionStitch Model (SSM), a program in which
the vacuum is a face-centered-cubic (FCC) lattice carrying a CSS stabilizer code [11, 12, 13, 14]. Each
input is stated below with its status tag, following the practice of the series.
2 The model in brief
This section gives a reader outside the SSM literature everything the derivation uses. Full construc-
tions are in Refs. [11, 12, 13, 14].
The vacuum is a lattice.
The SSM models the vacuum as a face-centered-cubic (FCC) lattice of
nodes at the Planck scale. FCC is the densest packing of spheres in three dimensions. Each node
touches
K = 12
nearest neighbors. The distance between neighbors is the bond length
L
, a xed
multiple of the Planck length.
Bonds carry entanglement, and the lattice runs a code.
Each bond between neighbors is a
maximally entangled pair. On top of this network the model places a quantum error-correcting code
of CSS type: one parity check on the twelve bonds at each node, and one on the twelve edges of
each octahedral gap between nodes. On a periodic patch with
L
3
s
sites this gives a
[[3L
3
s
, 2L
3
s
+2, 3]]
code [11]. The empty vacuum satises every check.
Matter is a defect.
A particle is a aw in the crystallization of this lattice: a pattern the checks ag
but cannot repair. The companion papers derive particle properties, including the proton-to-electron
mass ratio, from counting the operations such defects force [12, 13].
Two lengths follow from the geometry.
First, the lattice has a hard oor on distance. Two
nodes cannot approach closer than the circumradius of the triangular face of the coordination shell,
r
min
= L/
√
3
. Closer approach breaks the network topology. The model calls this the metric wall.
Second, the bond length
L
itself is xed by entropy: the vacuum state's entanglement across a at
boundary is exactly one bit per
L
2
of area (Appendix A), and matching this to the Bekenstein
Hawking law [1, 2]
S = A/4ℓ
2
P
xes
L
in Planck units [14].
One tested consequence.
The same two lengths predict that center-of-mass superpositions of
objects heavier than about
13 µ
g start to deform, and fail entirely above about
23 µ
g [14]. The
heaviest object yet held in superposition, a
16.2 µ
g acoustic resonator mode [16], sits inside this
window, where the model permits coherence. This paper shows that the same two lengths also x
the GUP parameter. Figure 1 shows the geometry that carries both lengths.
3 The two inputs
Input 1: the minimal length.
In the SSM, two lattice nodes cannot approach closer than the
circumradius of the cuboctahedral triangular face,
r
min
=
L
√
3
,
(3)
where
L
is the bond length. This metric wall is a geometric exclusion limit: closer approach destroys
the network topology. It is derived from the lattice geometry in Ref. [14] and conrmed there by a
companion simulation, which nds a sharp exclusion-radius transition at
R
ex
= L/
√
3 ≃ 0.577 L
with
a wide stability plateau. [
derived, within the model
]
Input 2: the bond length.
The series calibrates
L
against the BekensteinHawking area law [1, 2]
with one bit of entropy per boundary cell; the published value is Eq. (4) [14]. The calibration is
2
L
{111}
{100}
(a)
L
r
min
= L/
√
3
(b)
Figure 1: The geometry behind the two inputs. Panel (a) is drawn in isometric projection; panel (b)
shows the triangle in true shape within its own plane. (a) The FCC conventional cell: corner and
face-center nodes, the bond length
L
between nearest neighbors (heavy segment), a
{100}
plane (red,
dashed), and a
{111}
plane (blue). All bonds have the same length
L
; the two planes dier only in
how they cut the bond network, which is why their entanglement densities dier (Appendix A). (b)
The triangular face of the coordination shell, an equilateral triangle of side
L
lying in a
{111}
plane.
Its circumradius is the metric wall
r
min
= L/
√
3
, the minimal length of Input 1; its vertices are the
endpoints of the tripod bonds that cross a
{111}
cut.
not a convention. The exact entanglement entropy of the vacuum codeword of the
[[3L
3
s
, 2L
3
s
+2, 3]]
code family, computed across planar cuts at two lattice sizes, obeys a clean area law with exactly
one bit per
L
2
of
{100}
boundary area. The result is codeword-independent: the natural vacuum
choices are related by transversal Hadamard maps and lattice self-duality, which are local unitaries
and cannot change entanglement. The computation is given in Appendix A. Matching one bit per
L
2
to
S = A/4ℓ
2
P
gives
L = 2
√
ln 2 ℓ
P
= 1.6651 ℓ
P
.
(4)
[
derived; calibration facet selected empirically, see Section 5
]
4 The result and the bridge identity
The value.
Identify the operational minimum
∆x
min
of Eq. (1) with the metric wall of Eq. (3).
Then
β
0
=
r
min
ℓ
P
2
=
L
2
3 ℓ
2
P
=
4
3
ln 2 = 0.92420 . . .
(5)
The number is exact, closed-form, and parameter-free. It sits in the order-one range that quantum-
gravity arguments anticipate [5, 7]. [
derived, given the identication; see Section 5
]
The bridge identity.
The same two inputs x the SSM's prediction of a macroscopic decoherence
threshold: center-of-mass superpositions deform above
m
soft
= ℏ/(Lc) = 13.1 µ
g and fail above
m
hard
=
√
3 m
soft
= 22.6 µ
g [14]. Eliminating
L
between
m
soft
and
β
0
gives
β
0
=
1
3
m
P
m
soft
2
equivalently
m
soft
=
m
P
√
3β
0
.
(6)
3
Check:
1
3
(2.176 × 10
−8
kg/1.307 × 10
−8
kg)
2
= 0.924
. Equation (6) states that the GUP parameter
and the decoherence threshold are one measurement. A mass scan that locates
m
soft
current bulk
acoustic-wave resonators sit within a factor of two of the window [16] xes
β
0
. A laboratory
determination of
β
0
xes
m
soft
. The relation can be tested by anyone, including a reader who rejects
the model that produced it: it is a falsiable link between two data sets. [
derived, given the inputs
]
Comparison with bounds.
Table 1 places the prediction against representative experimental
limits. The prediction is consistent with all of them, and far below all of them. That is the expected
situation for any order-one
β
0
; the near-term test of this paper is Eq. (6), not a direct commutator
measurement.
Table 1: The predicted value against representative experimental upper bounds on
β
0
. Bound mag-
nitudes depend on the system and on how the deformation is assumed to act on composite objects;
see the cited papers.
Source Type Constraint on
β
0
This work closed-form prediction
=
4
3
ln 2 ≈ 0.924
Macroscopic oscillators [10] direct commutator test upper bounds,
≳ 10
6
scale
AURIGA bar detector [9] macroscopic mode upper bound, much larger
Quantum-system corrections [8] spectroscopy-type upper bounds, much larger
5 Assumptions and caveats
The identication is a stated step.
Equating the lattice exclusion radius with the operational
∆x
min
of Eq. (1) is physically natural: both are statements that no probe resolves structure below
that length. It is not a theorem. We tag it [
assumed
]. Dierent GUP deformations also relate
∆x
min
to
β
0
with dierent order-one factors; the value in Eq. (2) is stated in the quadratic convention of
Eq. (1) [6].
Two published calibrations, adjudicated.
The series carries two published values of
L
. The
decoherence paper uses one bit per
L
2
of boundary area, giving
L = 1.665 ℓ
P
[14]. The black-hole
paper instead postulates one ebit per
(111)
bilayer cell of area
p
2/3 L
2
, giving
L
0
≃ 1.843 ℓ
P
, and tags
that step plainly as a postulate used twice [15]. The exact computation of Appendix A discriminates
between them: the actual entanglement of the code state is one bit per
L
2
on
{100}
, and no facet of
the state carries the bilayer-cell density. The
{100}
value is therefore used here. One nontrivial cross-
check supports both papers: the black-hole paper's independent severed-bond count on the
(111)
plane,
2
√
3/L
2
0
, equals the exact
{111}
entanglement density of Appendix A the two computations
agree exactly where they measure the same quantity. Under the black-hole paper's bilayer postulate
the prediction would read
β
0
=
4
3
ln 2
p
3/2 ≃ 1.132
; we record it for completeness.
The calibration facet.
The vacuum's entanglement density is facet-dependent: one bit per
L
2
on
{100}
,
2
√
3
bits per
L
2
on
{111}
. Within the exact computation the facet must still be chosen,
and two independent grounds select
{100}
. First, the model's own calibration postulate is of Ryu
Takayanagi type: entropy is read on the
minimal
surface. On an anisotropic lattice a minimal cut
facets onto the cheapest plane, and
{100}
is the cheapest: a staircase of
{100}
microfacets undercuts
a direct
{111}
cut (
√
3 ≈ 1.73
versus
2
√
3 ≈ 3.46
bits per
L
2
) and a direct
{110}
cut (
√
2
versus
2
√
2
). The minimal-surface prescription therefore selects
{100}
by itself. Second, experiment agrees:
a
{111}
calibration would give
L = 3.099 ℓ
P
, move the decoherence window to
7.0
12.2 µ
g, and place
the observed coherent
16.2 µ
g cat state of Ref. [16] above the hard cuto, which is excluded. Under
4
that calibration the prediction would read
β
0
=
8
√
3
3
ln 2 ≃ 3.20
; we record it for completeness. The
two facets keep distinct roles in the model: matter defects and their thresholds live on
{111}
geometry,
while minimal entropy surfaces facet on
{100}
. One residual question is agged: staircase faceting of
a curved horizon carries a shape factor between faceted and smooth area, and resolving it is a dened
computation left to the series.
Composite systems.
Laboratory bounds on
β
0
from center-of-mass variables of composite oscil-
lators depend on how the deformation scales with constituent number, a known subtlety [10, 9].
Equation (6) inherits the same question through the model's total-mass coupling, which in the SSM
follows from additivity of rest energy under the model's scheduling mechanism. We ag the depen-
dence rather than hide it.
6 Conclusion
Two published inputs of the SSM a geometric minimal length and an entanglement-calibrated
bond length x the generalized uncertainty parameter in closed form:
β
0
=
4
3
ln 2 ≈ 0.924
. The
same inputs x the model's microgram decoherence window, so the two predictions collapse into one
identity,
β
0
=
1
3
(m
P
/m
soft
)
2
. One number, two experimental programs. Either program can now test
the other.
Declaration of competing interest
The author declares that he has no known competing nancial interests or personal relationships that
could have appeared to inuence the work reported in this paper.
Data availability
The entanglement computation of Appendix A is fully specied there and reproducible from the
stated construction; code implementing it is available from the author on request.
References
[1] J. D. Bekenstein, Black holes and entropy,
Phys. Rev. D
7
, 2333 (1973).
doi:10.1103/PhysRevD.7.2333.
[2] S. W. Hawking, Particle creation by black holes,
Commun. Math. Phys.
43
, 199 (1975).
doi:10.1007/BF02345020.
[3] D. Amati, M. Ciafaloni, and G. Veneziano, Can spacetime be probed below the string size?,
Phys. Lett. B
216
, 41 (1989). doi:10.1016/0370-2693(89)91366-X.
[4] M. Maggiore, A generalized uncertainty principle in quantum gravity,
Phys. Lett. B
304
, 65
(1993). doi:10.1016/0370-2693(93)91401-8.
[5] L. J. Garay, Quantum gravity and minimum length,
Int. J. Mod. Phys. A
10
, 145 (1995).
doi:10.1142/S0217751X95000085.
[6] A. Kempf, G. Mangano, and R. B. Mann, Hilbert space representation of the minimal length
uncertainty relation,
Phys. Rev. D
52
, 1108 (1995). doi:10.1103/PhysRevD.52.1108.
[7] F. Scardigli, Generalized uncertainty principle in quantum gravity from micro-black hole
gedanken experiment,
Phys. Lett. B
452
, 39 (1999). doi:10.1016/S0370-2693(99)00167-7.
5
[8] S. Das and E. C. Vagenas, Universality of quantum gravity corrections,
Phys. Rev. Lett.
101
,
221301 (2008). doi:10.1103/PhysRevLett.101.221301.
[9] F. Marin
et al.
, Gravitational bar detectors set limits to Planck-scale physics on macroscopic
variables,
Nat. Phys.
9
, 71 (2013). doi:10.1038/nphys2503.
[10] M. Bawaj
et al.
, Probing deformed commutators with macroscopic harmonic oscillators,
Nat.
Commun.
6
, 7503 (2015). doi:10.1038/ncomms8503.
[11] R. Kulkarni, A 67%-rate CSS code on the FCC lattice:
[[192, 130, 3]]
from weight-12 stabilizers,
arXiv:2603.20294 (2026).
[12] R. Kulkarni, Matter as incomplete crystallization,
Phys. Open
27
, 100423 (2026).
doi:10.1016/j.physo.2026.100423.
[13] R. Kulkarni, The massenergyinformation equivalence: a bottom-up identication
of the particle spectrum via FCC lattice error correction,
Phys. Open
(2026).
doi:10.1016/j.physo.2026.100414.
[14] R. Kulkarni, A two-step quantumclassical threshold at 13.122.6
µ
g with exact ratio
√
3
from
a close-packed vacuum lattice,
Quantum Rep.
8
(3), 78 (2026). doi:10.3390/quantum8030078.
[15] R. Kulkarni, Black holes in the FCC selectionstitch model: BekensteinHawking entropy, geo-
metric evaporation, and primordial-black-hole signatures,
Eur. Phys. J. Plus
141
, 916 (2026).
doi:10.1140/epjp/s13360-026-08148-9.
[16] M. Bild
et al.
, Schrödinger cat states of a 16-microgram mechanical oscillator,
Science
380
,
274 (2023). doi:10.1126/science.adf7553.
A The entanglement calibration
Setup.
Take the code family of Ref. [11] on a periodic FCC patch: qubits on the
3L
3
s
nearest-
neighbor bonds, one weight-12 check at each of the
L
3
s
/2
nodes, one weight-12 check on each of the
L
3
s
/2
octahedral voids. The vacuum is the codeword
|ψ⟩ ∝
Q
o
(1 + X
o
)|0 ···0⟩
. For a stabilizer state,
the entanglement of a region
A
is
S
A
= |A| − dim{
stabilizer elements supported inside
A}
, a GF(2)
rank computation.
Result.
For slab regions bounded by two at cuts, at both
L
s
= 4
(
[[192, 130, 3]]
) and
L
s
= 6
(
[[648, 434, 3]]
):
facet
S
slab
(bits, exact) bits per
L
2
per plane implied
L/ℓ
P
{100} L
2
s
− 1 1 2
√
ln 2 = 1.665
{110} 2L
2
s
− 1 2
√
2 2.800
{111} 2L
2
s
− 1 2
√
3 3.099
The slab formulas are exact at both sizes. The entanglement obeys a clean area law with a facet-
dependent coecient. The bond length itself is the same everywhere; what varies with the facet is the
areal geometry of the cut. Dierent planes sever dierent numbers of bonds per unit area (4,
3
√
2
,
and
2
√
3
per
L
2
on
{100}
,
{110}
, and
{111}
), just as surface energy in an ordinary crystal depends
on the Miller indices of the cleavage plane, and the checks that straddle each cut then constrain those
crossings by dierent amounts: on
{100}
only one independent bit per
L
2
survives, while on
{111}
the severed bonds are the tripod patterns of the code's logical operators, which no check constrains,
6
and the entropy saturates the crossing count. The nal column is a calibration exercise the value
of
L
each facet would imply if its density were matched to
S = A/4ℓ
2
P
not a statement that the
lattice spacing varies.
Codeword independence.
Three natural vacua were compared: the state above; the state with
vertex and octahedral check roles exchanged; and the dual state
C = ker(H
Z
)
with every logical
qubit in
|
¯
+⟩
. All three give identical entropies on every facet at both sizes. This is forced: the dual
state is the transversal-Hadamard image of its partner, local unitaries preserve entanglement, the two
check layouts map onto each other under the vertexoctahedron self-duality of FCC, and logical Pauli
frames are product unitaries. The coecients are properties of the code, not of a chosen codeword.
Facet selection.
The minimal-surface argument of Section 5 selects
{100}
: a staircase of
{100}
microfacets undercuts any direct cut on the other facets. The
{100}
coecient of one bit per
L
2
then reproduces the published calibration of Ref. [14], Eq. (4), upgrading it from convention to
computation.
7
